This invention relates generally to imaging systems, and more particularly, to controlling image reconstruction in imaging systems.
Imaging systems typically acquire scan data using a scanning operation and then reconstruct the image using the acquired scan data. Further, imaging systems perform image reconstruction using various methods, such as, for example, statistical methods. One such statistical method for image reconstruction includes the selection of a cost function η(x). The method further may include the application of a suitable iterative algorithm to determine a minimizer {circumflex over (x)} of the cost function η(x). The minimizer {circumflex over (x)} is a reconstructed image and may be defined by the following equation:{circumflex over (x)}=arg min η(x),  (1)where x is an image and is represented mathematically as x=(x1, x2, . . . xM) and M is the number of unknown pixel values (or voxel values in 3D).
Image reconstruction involves recovering an unknown function f(r) from the acquired scan data, where r denotes spatial position in 2D or 3D coordinates. Typically, a discretized version of f(r) is reconstructed. Often f(r) is represented using a finite-series expansion as follows:
                                          f            ⁡                          (              r              )                                =                                    ∑                              j                =                1                            M                        ⁢                                          x                j                            ⁢                                                b                  j                                ⁡                                  (                  r                  )                                                                    ,                            (        2        )            where bj(r) denotes spatial basis functions and each xj denotes an unknown coefficient. More generally, the function and bases can depend on time as well:
                              f          ⁡                      (                          r              ,              t                        )                          =                              ∑                          j              =              1                        M                    ⁢                                    x              j                        ⁢                                          b                j                            ⁡                              (                                  r                  ,                  t                                )                                                                        (        3        )            Therefore, determining f(r) simplifies to determining the coefficients, x. The spatial basis functions, b(r), may be selected to be, for example, rectangular functions, in which case, the coefficients x are called pixel or voxel values. Hereinafter, the term pixel values refers to any such set of coefficients, x, regardless of the choice of basis functions, b.
The cost function η(x) may be defined to include the following two terms:η(x)=L(x)+R(x)  (4)
The first term L(x) is typically referred to as the ‘data fit’ term, and is a measure of how well the image x fits the acquired scan data according to the physics, geometry, and statistics of the acquired scan data. The second term R(x) is typically referred to as the ‘regularization term’ or ‘roughness penalty’. R(x) controls noise, and without R(x), the minimum value {circumflex over (x)} becomes noticeably noisy.
Known methods typically use a regularization term that is a quadratic function of the differences between neighboring pixel values. The quadratic regularization term may cause blurring of edges, and loss of details (e.g., lower resolution) in the reconstructed image {circumflex over (x)} as a result of the value of the quadratic function rising rapidly. This blurring or loss of detail results from the large differences between neighboring pixel values. For example, there are large differences in pixel values for neighboring pixels that cross the boundaries between different anatomical regions and it is desirable to preserve the edges and other fine details in such a case, as well as reduce noise in the smoother image regions.
Known methods also may use another regularization term as follows:
                              R          ⁡                      (            x            )                          =                  β          ⁢                                    ∑                              j                =                1                            M                        ⁢                                          ∑                                  k                  ∈                                      N                    j                                                              ⁢                              Ψ                ⁡                                  (                                                            x                      j                                        -                                          x                      k                                                        )                                                                                        (        5        )            where Ψ is a potential function and is non-quadratic. Regularization terms of this form are referred to as edge-preserving regularization terms. Potential function Ψ may be computed, for example, using a Huber function or a hyperbola function as follows:
                              Ψ          ⁡                      (            t            )                          =                              (                          1              +                                                (                                      t                    /                    δ                                    )                                2                                                                        (        6        )            
The edge preserving potential functions rise less rapidly than the quadratic function, and hence better image quality is provided when compared to a regularization term having a quadratic function. The user selectable regularization parameter δ also controls the image contrast above which edges are preserved. However, these known methods use a single value of δ throughout the image, which can reduce image resolution and increase image noise, thereby reducing image quality.